We study harmonic maps between two distinct compact Riemann surfaces of the same genus. Our aim is to examine the role of the curvature in the relationship between energies of harmonic maps and conformal metrics on the target surface. We consider the energy of the energy minimizing map between the surfaces as a function of the metrics within the conformal class of the target and solve the variational problem of minimizing this functional when we restrict the upper curvature bound and normalize the area of the metrics. In doing so, we show that the upper curvature bound on the normalized target metric is an obstruction in obtaining a degree 1 conformal harmonic map between distinct Riemann surfaces.


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pp. 451-481
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